This paper focuses on continuous data assimilation (CDA) for the Navier-Stokes equations with nonlinear slip boundary conditions. CDA methods are typically employed to recover the original system when initial data or viscosity coefficients are unknown, by incorporating a feedback control term generated by observational data over a time period. In this study, based on a regularized form derived from the variational inequalities of the Navier-Stokes equations with nonlinear slip boundary conditions, we first investigate the classical CDA problem when initial data is absent. After establishing the existence, uniqueness and regularity of the solution, we prove its exponential convergence with respect to the time. Additionally, we extend the CDA to address the problem of missing viscosity coefficients and analyze its convergence order, too. Furthermore, utilizing the predictive capabilities of partial evolutionary tensor neural networks (pETNNs) for time-dependent problems, we propose a novel CDA by replacing observational data with predictions got by pETNNs. Compared with the classical CDA, the new one can achieve similar approximation accuracy but need much less computational cost. Some numerical experiments are presented, which not only validate the theoretical results, but also demonstrate the efficiency of the CDA.

翻译：本文重点研究具有非线性滑移边界条件的Navier-Stokes方程的连续数据同化问题。当初始数据或粘性系数未知时，连续数据同化方法通常通过引入由一段时间内观测数据生成的反馈控制项来恢复原始系统。在本研究中，基于从具有非线性滑移边界条件的Navier-Stokes方程的变分不等式导出的正则化形式，我们首先研究了初始数据缺失时的经典连续数据同化问题。在建立了解的存在性、唯一性和正则性之后，我们证明了其关于时间的指数收敛性。此外，我们将连续数据同化推广到处理粘性系数缺失的问题，并分析了其收敛阶。进一步地，利用部分演化张量神经网络对时间依赖问题的预测能力，我们提出了一种新颖的连续数据同化方法，即用pETNNs的预测结果替代观测数据。与经典连续数据同化相比，新方法能达到相近的逼近精度，但所需计算成本显著降低。文中给出了一些数值实验，不仅验证了理论结果，也证明了该连续数据同化方法的有效性。